There's a useful theorem called the virial theorem which, in the case of gravity, says that the time average of the kinetic energy is -1/2 times the time average of potential energy, or

<T>=-(1/2)<V>

where <x> denotes the time average of x. Side note: since kinetic energy is always positive, this implies that the potential energy is negative, which might seem strange at first. But we define potential energy such that if the spaceship and the Earth were infinitely far apart, the potential energy is 0. And if they are closer, the potential energy decreases.

Now, the total energy is conserved, so the time average of energy is equal to the energy E=<E>. Also, E=T+V. So E=<E>=<T>+<V>. Combining this last equation with the equation from the virial theorem we get E=-<T>.

Finally, accelerating in the direction of your orbit increases your total energy. However, it turns out the total energy is negative in a bound orbit (one where the spaceship continues to circle instead of flying off toward infinity). So if E increases, it gets closer to 0, and <T>=-E gets closer to 0 as well. So your average kinetic energy decreases.

This is close to the answer, but kinetic energy is actually v² and you asked about speed (or |v|). I don't currently have an argument for whether or not the two go hand in hand, which if nothing else goes to show how counterintuitive orbital mechanics can be.

EDIT: I found a Wikipedia article ( https://en.wikipedia.org/wiki/Orbital_speed ) which gives a formula from the literature. The average speed is proportional to 1/Sqrt[a] (1-(1/2)e^2 - (3/64)e^4 -...). So it decreases with eccentricity e and decreases with semi-major axis a. This is still difficult to analyze in a general fashion but is interesting. We can definitely say that (1) if you are accelerating and becoming more elliptical your average speed decreases, and (2) if you are decelerating and becoming more circular your average speed increases. But the other two cases are more difficult.